Name
MATH 107 PRACTICE
MIDTERM III Please work out each of the given problems.
Credit will be based on the steps that you show towards the final answer.
Show your work. PROBLEM 1 Please answer the following true or false.
If false, explain why or provide a counter example.
If true, explain why. A) Let f(x,y) be a
function of two variables and P be a point.
If the value of the function tends towards 5
for every line segment that ends at P, then
False, the limit has to be the same for all curves not just lines. For example, if P is the origin (0,0) and
xy2 + x2y Then it is easy to check that the limit is zero along all line segments ending at the origin, but along y = x2 the limit is 1/2.
B) If z
= f(x - y)
and f is a continuous function then
True. Use the chain rule with u = x - y: zx = f '(u)(ux) = f '(u)(1) = f '(u) zy = f '(u)(uy) = f '(u)(-1) = -f '(u) and f '(u) + (-f '(u)) = 0
PROBLEM 2 Find the dimensions of the rectangular box with the largest volume in the first octant such that one of the vertices is at the origin and the opposite vertex lies on the ellipsoid
x2 + 2y2
+ 3z2 = 6
The volume of such a box is given by V = xyz We use LaGrange Multipliers grad F = <2x,4y,6z> = l<yz,xz,xy> = l grad V This gives us four equations
2x = lyz
4y = lxz
6z = lxy
x2 + 2y2
+ 3z2 = 6
Solving the first three equations for l gives l = 2x/yz l = 4y/xz l = 6z/xy Setting the equations equal to each other and multiplying by xyz gives 2x2 = 4y2 = 6z2 or x2 = 2y2 = 3z2 Substituting into the fourth equation gives x2 + x2 + x2 = 6 x2 = 2
x = Resubstituting gives 2 = 2y2 y = 1 and 2 = 3z2 z = (2/3)0.5
PROBLEM 3 Consider the function
5 A.
Use a calculator to sketch the level curves corresponding to We solve for y
5
B.
Suppose that z represents the altitude
function, and you are to travel on this surface above the unit circle in a
counter-clockwise direction. Discuss
your travels.
Beginning at the point (1,0) we are infinitely
high. Then we slide steeply down, reaching a local minimum
elevation at the point (
PROBLEM 4 Find the following limits if they exist. A.
We consider the path to the origin from the positive x axis (y = 0). We get
x4 Now approach the origin via y = x. We have
x4 + 2x4 + x4
4x4 Since the two limits approach two different values, we can conclude that the limit does not exist.
B.
For this limit we use polar coordinates
r5cos5q - r4sin4q Therefore, the limit exists and is equal to 0
PROBLEM 5 Let
f(x,y)
= x2 + xy,
x(u,v) = u2v, and
y(u,v) = u - v
Find A)
fx = 2x + y
B) Without calculating any derivatives, write the appropriate chain rule for
fuu = (fu)u = (fxxu + fyyu)u = fxuxu + fxxuu + fyuyu + fyyuu = (fxxxu + fxyyu)xu + fxxuu + (fxyxu + fyyyu)yu + fyyuu
C)
(7
Points)Find
x(1,2) = 12(2) = 2 y(1,2) = 1 - 2 = -1 xu = 2uv = 2(1)(2) = 4 xuu = 2v = 2(2) = 4 yu = 1 yuu = 0 fxx = 2 fxy = 1 fyy = 0 fx = 2x + y = 2(2) + (-1) = 3 fy = x = 2 Plugging in, we get [(2)(4) + (1)(1)](4) + (3)(4) + [(1)(4) + (0)(1)](1) + (2)(0) = (9)(4) + 12 + (4)(1) = 36 + 12 + 4 = 52
PROBLEM 6 If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2, and R3, then
1
1
1
1
If the resistances are measured as R1
= 25 ohms, R2 = 40 ohms,
and R3 = 50 ohms, with possible errors
of 0.5% in each case, use differentials to estimate
the maximum error in the calculated value of R. We use the differential formula -R-2 dr = -R1-2 dR1 - R2-2 dR2 - R3-2 dR3 R-2 dr = .005(R1-2 R1 + R2-2 R2 + R3-2 R3) R-2 dr = .005(R1-1 + R2-1 + R3-1 ) = .005R Hence dr = .005R3 = .005(1/25 + 1/40 + 1/50)3 = .000425
PROBLEM 7 Find parametric equations for the tangent
line of the curve of intersection of the paraboloid
z = x2 + y2
and the ellipsoid 4x2 + y2 + z2
= 9
at the point (-1,1,2). This line is on the tangent plane of both surfaces. Thus is is perpendicular to both normal vectors. We compute the two normal vectors. The normal vector for the parabola is <2x, 2y, -1> = <-2, 2, -1> The normal vector for the ellipsoid is <8x, 2y, 2z> = <-8, 2, 4> Now cross the two vectors to find a vector perpendicular to both normal vectors.
10i + 16j + 12k Now use the formula for parametric equations of a line given a point and a parallel vector x(t) = -1 + 10t y(t) = 1 + 16t z(t) = 2 + 12t
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